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I 3x y =- fcruv -1shy

2 f ( 3x )

x -3

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7M o gt1 ETTLfl~KUVOlr = 8 - - 1 shy

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s

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H pom) Mpavdar iJTTOAOY[~CTO I IK TOO Tunou J = 4~E ~ lila TO lilaypoflflo iii 0

dK dy = mm

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Sw = W bull 1 0 545

P _ 9EJ ] aTlt Sp- Sb= [ Ii 41

Kat Sw = Sp - Sb

EK T~S ES = O xa ~- ~ 0 Aa~~aVO~(VTJln~D 755

AOAITIKWV 2EJ EJ1= 3 V xed I = w 87]52

opnXOVIKWV Po

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WS TO TOU AUyO~OU ~S ypa~~~s

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2 ) Ta liKpa poundLVCll ma8(pws TT(TTaKTw~tva

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Y TO ptAOS Ka~~(WS

M ~ pO~ Ka~~(WS

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H t~[OWOlS ~S ypa~~~s Ka~~(WS dVCll

1(1 2)Y=T -cruv - 1- x

lv8a aa y =0

Y =0

Y = r

768 Ivao lila I (2) 2x= o 2 I y= O IlTToU y = 2 - - - l1L - - X

r ( 2 ) 2Y = 2 - - auv - - x

EJ I ( 2 ) 2 KO M=- EJY =- 2 - - auv - 1- x

I H liuvalK~ tvtpyca KCtItlCWs Ab riVal lOT) TTpOS

- bull M Ab= f 2EJ dx = EJI T o

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bull Ab= f gydx

o

3 To fpyov TOU qopTlou S A8 rlva laov

A ~ S

H tTTlltlKuvaS Tiis ypoliliiiS 11 riVal -lOT) TIpes

1 bull bullt1 = 2 f ydx = I ~I o

H Ct~OVK~ IiUvali~ P riVal tOT) TTp6~ t1

P = E F - shy1

4 To lpyov Tiis ~ov~S liuvalipoundw~ Pv Iia ~v tTTllitlKUVOlV Tiis ypalili~~ Av rlva laoVTTp6S

5 To fpyov Tiis avomuaaolitv]~ 8AmKijs liuvCtlicwS Pp Ap riVal iaov TTp6~

f l 7t1

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MCTa~alvovTcS lit poundIS looppoTTlav liuvCtIipoundwV ixolipoundv litll

S = ampAb = 2EJI ~ b amp1 I

ampAb Sh=I= gT

ampAS8 = tif = S

~v KCtTWa ~Iawalv Sb + Sb+Sv= S +Sp

Kal

= I + ( - P 2fft) + EJ -()S g 2 2EJ T 8- -I _

757 lJl~1 IDlITIIWV 5LaDXOVIKWV x

T Sp f

LX 5 AUYlC1OC rile ypafJr ICOTa TO I(QTaK6pu+o tntm6oY

( ETTEVtpyouaal 6UV6~LC)

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OUTW

Kat TO PChOS f bull

213 ( S--f)

f = (hEJ - Pol)

A bullbull df 0oUla Imln u(OV (j[ =

Kat tav S = 0 ITpOKUmtl

Kat

f11Q D(DOflCVOV TPOITOV tmDofliis ~TOI g Kal J yvwara Kat D(DOfltvlV Dlacjgtopav 8(PflOKpashyalas f1t ijTOl P yvwarov haflpllvOfl(V nJV aUVlarwaav ouvafllV S WS auvapTlOlV TWV I Katl

OUTW ITaparWfl(V ypacjgtIKWS nJv WS Iivw auvapTlOlV xapaaaOVTts nJV ITpoundppnAhOUaaV TWV auvapnJapoundwv S = F ( r) (Dla Dlacjgtopa 1 )

OUTW DIll Tiis KaflITUhlS TauTlS tiS TOllS i~ovas auvTtTaYfltvwv SKat f Duvafl(eO va ITpoaDloplawflpoundV TO aITQlTOUflpoundVOV PthOS KOfltjJ(WS TO OIToiov tv auvDuaaflltlgt fit nJV aUVlshyarwaav ouvafllV S ouvaTOl v afToPil tmKlvDUVOV Dla nJv tmDofl~v flOS ~TOI va fTPOKahpound01J hUYlaflov Tiis ypoflfliis KOTa TO KaTOKOpucjgtOV poundfTlfTpoundDOV

H TOlOUTl aITpoundIKOVIOlS Tiis auvoPnlapoundws S = F (g J Po Imiddotf) tmTptfTpound1 nJV aUYKpav DlOcjgt6pwv TUITWV tmDofliis (DlOcjgtOpOl TUITOI OlDlpOTpOX~)v J DlOcjgtOpOl TUITOl arpwshynlPWV g) fIpoundTO~U TWV Kelt TOV IToplaflov aUfllTpoundpoaflOrwv tITt Tiis aacjgtahdos EKaarou TUTTOU

nplnTWal~ II)

IlDlPOTPOXla TUITOU S 49 J =2 x 1781 em E =21 X 1000kp em ITpwTijppounds ~UhlVOl gk 26 = P lem f1lQcjgtopa epoundPfloKpaalos f1t = 45deg C Dl fjv avomUaatTOl a~ovl~

8hlmlK~ ouvofllS Po = 140 Mp

7M o gt1 ETTLfl~KUVOlr = 8 - - 1 shy

AUTT] poundtval lila T~V apxIKWr EuScLav ypaflfl~v ~ m9avwTtpa floPcjgt~ napaflopcjgtWOEWr wr TOUTO oacjgtwr npOKumEI tK Tijr nElpar

Al KaflnUAQL Aafl~ovOVTQL w~ ~flITOVOElliELr lila TOV SEwpl)nKov unoAoYloflOV KaTa lidjTEPOV ~ oVTiOTaOIr dr tYKapOiav flETaK[VI)OiV WAafl~oVETai wr OTa9Epo KalToi TOUTO liEV OUflPaivEI liia Aiav fllKpa ~tAI) napaflopcjgtWOEWr wr npoK umu EK TWV liIEVEPshyYI]9ElOWV flETp~OEWV

s

bull rCE7

1= t-v~~ys I 0 t ~(I -w 7 xlll 1- i =- IOW dL

X= l y - 2 (I t 0 lfK-) - -I

IX 3 JuyIOKlC riic ypa~~~r KOTlI TO 6PL~61f10l trrincOol

Mop Tft YPQ~~fic 1(6~~pound~ aCWpT)nK~ OLt ptUVl0u ( 18 t Sp)

KaTa TplTov liEx6f1ESa on ~ toxopa (Oilil)poTpoXIai - OTpWTijpEr ) KOla T~V dr KOfl~IV napafl6pcjgtwoIV EXCI fllav OTaSCpaV pom)v olipavEiar

H oGTwr opa9ELoa pom) olipavdar liuvaTQt va unoAoyoSij nElpaflanKwr wr t~ijr H toxapa wr aflcjgtltpEIOTOr liOKOr tnt fI~KOUr L cjgtopTi~ETQt lila oUYKEVTpwfltvou cjgtopTiou K Eit TO fltoov xapaOOETai lit TO liloypaflfla flETapoAijr TWV PEAWV y ouvapTlloci TOO K

H pom) Mpavdar iJTTOAOY[~CTO I IK TOO Tunou J = 4~E ~ lila TO lilaypoflflo iii 0

dK dy = mm

Ilta TWV TplWV TOUTWV llTTAOnOI~oEwV unoAoyi~ETol CUKOAW~ ~ KOTOOTOOir aOTOSOOr ooppeniar

H KaTaOTaole aOTaSO Oe ooppon[or I EupioKClai nOXlv lila oUYKpioEWr TWV Incvcpyoushyowv liuvoflEWV (IX 4)

To aSpoiofla TWV liuvoflEWV aVTIOTOOEWe dr tyKapoiav flETaTOmOiV Sw Kat K6f1~IV Sb lito V va tlVQt ioov fit nlv IiUvafllv Sp poundK Tijr anEACUSEPOUfltVI]r tVEpytiar

llElioflt vou on al Sp Kat Sb flaSl]flanKwr poundIvaI Tije auTijr flO pcjgtijr Aafl~6vovTal ofloO AUTal au~6vouv ypaflfllKWr flCTIl TOG r lliloTunla de TO npopAl)fla Elval ~ aVTllipaOir dr nlv tyKapoiav flClaTomOiv W ~nr elval naSl)nK~

OOTW fila lilaT6pa~le Ete nlv Stolv Tijr ypaflflijr ~Tlr poundAOpt xwpav lK nvoe ahlae liEV t~ilAticjgttTal flovl) TI)e iiI Enavacjgtopoe Tijr ypaflflijr tlr nlv StOiv TT]e OAAa napafltvtl Oihw lillllicliofltvl)v pAamlK~v liuvafllv P lipwoav tnt Tie ypGflflijr unoAoYlo9tvTWV ntlshypaflanKwe TWV J Ka W EXOfltV

Sw = W bull 1 0 545

P _ 9EJ ] aTlt Sp- Sb= [ Ii 41

Kat Sw = Sp - Sb

EK T~S ES = O xa ~- ~ 0 Aa~~aVO~(VTJln~D 755

AOAITIKWV 2EJ EJ1= 3 V xed I = w 87]52

opnXOVIKWV Po

EK TOU alaypa~~aTOS CUVClTCll va yiv~ OUYKPatS TOU f ~pound uTTo8poundTIKOV oltjgtaA~a 8to(wS T~S ypa~~~s f aa T~V dp(atv T~S aoqaAdas ~S ypa~~~s

E1S T~V TTp1i~IV o~wS ~ UTTO 8AlmKas auva~poundlts (UPoKo~tV1l ypa~~~ ultjgtlmam T~V tTTIshyopaolv TWV KUKAOq0POUVTWV oup~wv ~ns o~wS apoundv EXpoundlt ~txP O~~(POV UTTOAoy08~ ~pound apK(Tilv aKpi~poundltav

Elva ~(~aiws YVWOTOV on 0 TTpO Kat ~poundTa TOV Oup~OV a~~IOUpyOU~(VOS Ku~ano~os ~S ypa~~~s EXpoundlt WS OTToTtApoundO~a T~V TTpOOWPv~v ~poundiwatv TWV OUV8~KWV TP~~S dS TeO lp~a

ETTi01]S dS T~V ~poundTa ~KPWV oqaA~aTwv 8to(wS TOTT08(TTWtV~V ypa~~~v ovamuooovm KaTa T~V KUKAoqopiav TTApoundupIKalauva~poundlts

Aihal m8avws va aVTlom8~iSwvm AIKWS ~ ~(PIKWS My TWV ~u~~~tvwv TP~WV OUV(shyTTd TOU qopTiou

Eltjgt OOOV Aomov uTTClpxouv oi poundK T~S KUKAoqopias auo~(v(iS TTapayoVT(S oiTlv(S opoundv ouvavTCll ~txP af]~(POV oaqws va poundKTI~~8wat atov OTTWS 0 lJTTOAoylo~oS TOU ltmAou AUyO~ou yivU KaTa TOOUTOV TPOTTOV WOTE ~ ypa~~~ va pound~aoqaAiS(ml Kal poundK TWV TTpoO-8poundTWV TOlJTWV KLVOUVWV

TOUTO CUVClTCll va poundTTlTEux8ij aa TWV S aaypa~~aTwv

H TTp1i~IS ~(~aiws oTToapoundltKvupoundlt on TTpoKumpoundlt AUyO~OS ~S ypa~~~s amv OU~TTtoouV TTApoundlmol 8(TlKWS ala TOV AUylo~oV poundTT(V(PYOUVT(S TTapayoVT(S

Olhws poundav KaTa TT(pioaov U~~A~S 8(p~OKpaoias xaAapw8~ ~ ypa~~~ ylvwvml poundpyaolOl OUVT~P~O(WS ds TO Ep~a altASu ai ala TOU poundpyom~iou oup~oS TpOXOTT(aOU~(voS ~t mUToxpoVOv aG~~OIv TOU p AUYIO~OS dVCll ovaTToqpounduKTOS

E1S Ta TTA(OV(Knj~aTa Aomov T~S ypa~~~s liv(u ap~wv OVTlTaOOOVTOI Kal ~poundlOV(Knj~aTa

WS TO TOU AUyO~OU ~S ypa~~~s

Iva oTTaAAayw~(V poundK TOUTOU TTptTTpoundl va ~PW~(V au~pws TaS Epyaolas OUV~P~O(WS

T~S ypa~~~s TTA~V TWV EmpaAAo~tvwv opWV EK TOU 8(WP~TlKOU UTTOAOYIO~OU

3 0poundWPl]TKi) bAuai TIii TTpoundpTTTwapoundwi TOU Auyaou TIii ypoIii KOTQ TO KOTOKOPU4gt0Y ETTiTTpound1oY

lila T~V 8(Wp~TlK~V ETTiAUOIV yivovm ai KaTw81 cuo paOlKal TTapaaoxal

I ) H ypa~~~ Ka~~(WS dVCll ~~ITOVO(la~s

2 ) Ta liKpa poundLVCll ma8(pws TT(TTaKTw~tva

-EoTwoav TO ~~KOS TWV 0I0~pOTpOXWV

g TO lalov papos T~S ypa~~~s

E TO ~tTpOV EAaOTlK6~TOS TOU XaAUPoS TWV OIa~pOTpOXIWV

J ~ pO~ aopavdas T~S ypa~~~s

Y TO ptAOS Ka~~(WS

M ~ pO~ Ka~~(WS

P ~ poundV Tij ypa~~fj ovamuooo~tv~ 8Alm1K~ cuva~s

H t~[OWOlS ~S ypa~~~s Ka~~(WS dVCll

1(1 2)Y=T -cruv - 1- x

lv8a aa y =0

Y =0

Y = r

768 Ivao lila I (2) 2x= o 2 I y= O IlTToU y = 2 - - - l1L - - X

r ( 2 ) 2Y = 2 - - auv - - x

EJ I ( 2 ) 2 KO M=- EJY =- 2 - - auv - 1- x

I H liuvalK~ tvtpyca KCtItlCWs Ab riVal lOT) TTpOS

- bull M Ab= f 2EJ dx = EJI T o

2 To lpyo-v TO CtTIaTOUf1poundVOV Iia ~V avutlwav Tiis ypall~S An TTpOS uTTcpvIK1]OIV TOU 16lou ~CtpOUS rlva

bull Ab= f gydx

o

3 To fpyov TOU qopTlou S A8 rlva laov

A ~ S

H tTTlltlKuvaS Tiis ypoliliiiS 11 riVal -lOT) TIpes

1 bull bullt1 = 2 f ydx = I ~I o

H Ct~OVK~ IiUvali~ P riVal tOT) TTp6~ t1

P = E F - shy1

4 To lpyov Tiis ~ov~S liuvalipoundw~ Pv Iia ~v tTTllitlKUVOlV Tiis ypalili~~ Av rlva laoVTTp6S

5 To fpyov Tiis avomuaaolitv]~ 8AmKijs liuvCtlicwS Pp Ap riVal iaov TTp6~

f l 7t1

Ap = Pot1 = PO 4T

MCTa~alvovTcS lit poundIS looppoTTlav liuvCtIipoundwV ixolipoundv litll

S = ampAb = 2EJI ~ b amp1 I

ampAb Sh=I= gT

ampAS8 = tif = S

~v KCtTWa ~Iawalv Sb + Sb+Sv= S +Sp

Kal

= I + ( - P 2fft) + EJ -()S g 2 2EJ T 8- -I _

757 lJl~1 IDlITIIWV 5LaDXOVIKWV x

T Sp f

LX 5 AUYlC1OC rile ypafJr ICOTa TO I(QTaK6pu+o tntm6oY

( ETTEVtpyouaal 6UV6~LC)

f1pound1iofUvoU on ( rII ) t1VQl fllKpOV TO (f II ) WS ClTTelpOaTOV avwTtpas Tll~(wS tmTptshyITel nJV ITapllMlIVIV TOU opOU Sv

OUTW

Kat TO PChOS f bull

213 ( S--f)

f = (hEJ - Pol)

A bullbull df 0oUla Imln u(OV (j[ =

Kat tav S = 0 ITpOKUmtl

Kat

f11Q D(DOflCVOV TPOITOV tmDofliis ~TOI g Kal J yvwara Kat D(DOfltvlV Dlacjgtopav 8(PflOKpashyalas f1t ijTOl P yvwarov haflpllvOfl(V nJV aUVlarwaav ouvafllV S WS auvapTlOlV TWV I Katl

OUTW ITaparWfl(V ypacjgtIKWS nJv WS Iivw auvapTlOlV xapaaaOVTts nJV ITpoundppnAhOUaaV TWV auvapnJapoundwv S = F ( r) (Dla Dlacjgtopa 1 )

OUTW DIll Tiis KaflITUhlS TauTlS tiS TOllS i~ovas auvTtTaYfltvwv SKat f Duvafl(eO va ITpoaDloplawflpoundV TO aITQlTOUflpoundVOV PthOS KOfltjJ(WS TO OIToiov tv auvDuaaflltlgt fit nJV aUVlshyarwaav ouvafllV S ouvaTOl v afToPil tmKlvDUVOV Dla nJv tmDofl~v flOS ~TOI va fTPOKahpound01J hUYlaflov Tiis ypoflfliis KOTa TO KaTOKOpucjgtOV poundfTlfTpoundDOV

H TOlOUTl aITpoundIKOVIOlS Tiis auvoPnlapoundws S = F (g J Po Imiddotf) tmTptfTpound1 nJV aUYKpav DlOcjgt6pwv TUITWV tmDofliis (DlOcjgtOpOl TUITOI OlDlpOTpOX~)v J DlOcjgtOpOl TUITOl arpwshynlPWV g) fIpoundTO~U TWV Kelt TOV IToplaflov aUfllTpoundpoaflOrwv tITt Tiis aacjgtahdos EKaarou TUTTOU

nplnTWal~ II)

IlDlPOTPOXla TUITOU S 49 J =2 x 1781 em E =21 X 1000kp em ITpwTijppounds ~UhlVOl gk 26 = P lem f1lQcjgtopa epoundPfloKpaalos f1t = 45deg C Dl fjv avomUaatTOl a~ovl~

8hlmlK~ ouvofllS Po = 140 Mp

EK T~S ES = O xa ~- ~ 0 Aa~~aVO~(VTJln~D 755

AOAITIKWV 2EJ EJ1= 3 V xed I = w 87]52

opnXOVIKWV Po

EK TOU alaypa~~aTOS CUVClTCll va yiv~ OUYKPatS TOU f ~pound uTTo8poundTIKOV oltjgtaA~a 8to(wS T~S ypa~~~s f aa T~V dp(atv T~S aoqaAdas ~S ypa~~~s

E1S T~V TTp1i~IV o~wS ~ UTTO 8AlmKas auva~poundlts (UPoKo~tV1l ypa~~~ ultjgtlmam T~V tTTIshyopaolv TWV KUKAOq0POUVTWV oup~wv ~ns o~wS apoundv EXpoundlt ~txP O~~(POV UTTOAoy08~ ~pound apK(Tilv aKpi~poundltav

Elva ~(~aiws YVWOTOV on 0 TTpO Kat ~poundTa TOV Oup~OV a~~IOUpyOU~(VOS Ku~ano~os ~S ypa~~~s EXpoundlt WS OTToTtApoundO~a T~V TTpOOWPv~v ~poundiwatv TWV OUV8~KWV TP~~S dS TeO lp~a

ETTi01]S dS T~V ~poundTa ~KPWV oqaA~aTwv 8to(wS TOTT08(TTWtV~V ypa~~~v ovamuooovm KaTa T~V KUKAoqopiav TTApoundupIKalauva~poundlts

Aihal m8avws va aVTlom8~iSwvm AIKWS ~ ~(PIKWS My TWV ~u~~~tvwv TP~WV OUV(shyTTd TOU qopTiou

Eltjgt OOOV Aomov uTTClpxouv oi poundK T~S KUKAoqopias auo~(v(iS TTapayoVT(S oiTlv(S opoundv ouvavTCll ~txP af]~(POV oaqws va poundKTI~~8wat atov OTTWS 0 lJTTOAoylo~oS TOU ltmAou AUyO~ou yivU KaTa TOOUTOV TPOTTOV WOTE ~ ypa~~~ va pound~aoqaAiS(ml Kal poundK TWV TTpoO-8poundTWV TOlJTWV KLVOUVWV

TOUTO CUVClTCll va poundTTlTEux8ij aa TWV S aaypa~~aTwv

H TTp1i~IS ~(~aiws oTToapoundltKvupoundlt on TTpoKumpoundlt AUyO~OS ~S ypa~~~s amv OU~TTtoouV TTApoundlmol 8(TlKWS ala TOV AUylo~oV poundTT(V(PYOUVT(S TTapayoVT(S

Olhws poundav KaTa TT(pioaov U~~A~S 8(p~OKpaoias xaAapw8~ ~ ypa~~~ ylvwvml poundpyaolOl OUVT~P~O(WS ds TO Ep~a altASu ai ala TOU poundpyom~iou oup~oS TpOXOTT(aOU~(voS ~t mUToxpoVOv aG~~OIv TOU p AUYIO~OS dVCll ovaTToqpounduKTOS

E1S Ta TTA(OV(Knj~aTa Aomov T~S ypa~~~s liv(u ap~wv OVTlTaOOOVTOI Kal ~poundlOV(Knj~aTa

WS TO TOU AUyO~OU ~S ypa~~~s

Iva oTTaAAayw~(V poundK TOUTOU TTptTTpoundl va ~PW~(V au~pws TaS Epyaolas OUV~P~O(WS

T~S ypa~~~s TTA~V TWV EmpaAAo~tvwv opWV EK TOU 8(WP~TlKOU UTTOAOYIO~OU

3 0poundWPl]TKi) bAuai TIii TTpoundpTTTwapoundwi TOU Auyaou TIii ypoIii KOTQ TO KOTOKOPU4gt0Y ETTiTTpound1oY

lila T~V 8(Wp~TlK~V ETTiAUOIV yivovm ai KaTw81 cuo paOlKal TTapaaoxal

I ) H ypa~~~ Ka~~(WS dVCll ~~ITOVO(la~s

2 ) Ta liKpa poundLVCll ma8(pws TT(TTaKTw~tva

-EoTwoav TO ~~KOS TWV 0I0~pOTpOXWV

g TO lalov papos T~S ypa~~~s

E TO ~tTpOV EAaOTlK6~TOS TOU XaAUPoS TWV OIa~pOTpOXIWV

J ~ pO~ aopavdas T~S ypa~~~s

Y TO ptAOS Ka~~(WS

M ~ pO~ Ka~~(WS

P ~ poundV Tij ypa~~fj ovamuooo~tv~ 8Alm1K~ cuva~s

H t~[OWOlS ~S ypa~~~s Ka~~(WS dVCll

1(1 2)Y=T -cruv - 1- x

lv8a aa y =0

Y =0

Y = r

768 Ivao lila I (2) 2x= o 2 I y= O IlTToU y = 2 - - - l1L - - X

r ( 2 ) 2Y = 2 - - auv - - x

EJ I ( 2 ) 2 KO M=- EJY =- 2 - - auv - 1- x

I H liuvalK~ tvtpyca KCtItlCWs Ab riVal lOT) TTpOS

- bull M Ab= f 2EJ dx = EJI T o

2 To lpyo-v TO CtTIaTOUf1poundVOV Iia ~V avutlwav Tiis ypall~S An TTpOS uTTcpvIK1]OIV TOU 16lou ~CtpOUS rlva

bull Ab= f gydx

o

3 To fpyov TOU qopTlou S A8 rlva laov

A ~ S

H tTTlltlKuvaS Tiis ypoliliiiS 11 riVal -lOT) TIpes

1 bull bullt1 = 2 f ydx = I ~I o

H Ct~OVK~ IiUvali~ P riVal tOT) TTp6~ t1

P = E F - shy1

4 To lpyov Tiis ~ov~S liuvalipoundw~ Pv Iia ~v tTTllitlKUVOlV Tiis ypalili~~ Av rlva laoVTTp6S

5 To fpyov Tiis avomuaaolitv]~ 8AmKijs liuvCtlicwS Pp Ap riVal iaov TTp6~

f l 7t1

Ap = Pot1 = PO 4T

MCTa~alvovTcS lit poundIS looppoTTlav liuvCtIipoundwV ixolipoundv litll

S = ampAb = 2EJI ~ b amp1 I

ampAb Sh=I= gT

ampAS8 = tif = S

~v KCtTWa ~Iawalv Sb + Sb+Sv= S +Sp

Kal

= I + ( - P 2fft) + EJ -()S g 2 2EJ T 8- -I _

757 lJl~1 IDlITIIWV 5LaDXOVIKWV x

T Sp f

LX 5 AUYlC1OC rile ypafJr ICOTa TO I(QTaK6pu+o tntm6oY

( ETTEVtpyouaal 6UV6~LC)

f1pound1iofUvoU on ( rII ) t1VQl fllKpOV TO (f II ) WS ClTTelpOaTOV avwTtpas Tll~(wS tmTptshyITel nJV ITapllMlIVIV TOU opOU Sv

OUTW

Kat TO PChOS f bull

213 ( S--f)

f = (hEJ - Pol)

A bullbull df 0oUla Imln u(OV (j[ =

Kat tav S = 0 ITpOKUmtl

Kat

f11Q D(DOflCVOV TPOITOV tmDofliis ~TOI g Kal J yvwara Kat D(DOfltvlV Dlacjgtopav 8(PflOKpashyalas f1t ijTOl P yvwarov haflpllvOfl(V nJV aUVlarwaav ouvafllV S WS auvapTlOlV TWV I Katl

OUTW ITaparWfl(V ypacjgtIKWS nJv WS Iivw auvapTlOlV xapaaaOVTts nJV ITpoundppnAhOUaaV TWV auvapnJapoundwv S = F ( r) (Dla Dlacjgtopa 1 )

OUTW DIll Tiis KaflITUhlS TauTlS tiS TOllS i~ovas auvTtTaYfltvwv SKat f Duvafl(eO va ITpoaDloplawflpoundV TO aITQlTOUflpoundVOV PthOS KOfltjJ(WS TO OIToiov tv auvDuaaflltlgt fit nJV aUVlshyarwaav ouvafllV S ouvaTOl v afToPil tmKlvDUVOV Dla nJv tmDofl~v flOS ~TOI va fTPOKahpound01J hUYlaflov Tiis ypoflfliis KOTa TO KaTOKOpucjgtOV poundfTlfTpoundDOV

H TOlOUTl aITpoundIKOVIOlS Tiis auvoPnlapoundws S = F (g J Po Imiddotf) tmTptfTpound1 nJV aUYKpav DlOcjgt6pwv TUITWV tmDofliis (DlOcjgtOpOl TUITOI OlDlpOTpOX~)v J DlOcjgtOpOl TUITOl arpwshynlPWV g) fIpoundTO~U TWV Kelt TOV IToplaflov aUfllTpoundpoaflOrwv tITt Tiis aacjgtahdos EKaarou TUTTOU

nplnTWal~ II)

IlDlPOTPOXla TUITOU S 49 J =2 x 1781 em E =21 X 1000kp em ITpwTijppounds ~UhlVOl gk 26 = P lem f1lQcjgtopa epoundPfloKpaalos f1t = 45deg C Dl fjv avomUaatTOl a~ovl~

8hlmlK~ ouvofllS Po = 140 Mp

768 Ivao lila I (2) 2x= o 2 I y= O IlTToU y = 2 - - - l1L - - X

r ( 2 ) 2Y = 2 - - auv - - x

EJ I ( 2 ) 2 KO M=- EJY =- 2 - - auv - 1- x

I H liuvalK~ tvtpyca KCtItlCWs Ab riVal lOT) TTpOS

- bull M Ab= f 2EJ dx = EJI T o

2 To lpyo-v TO CtTIaTOUf1poundVOV Iia ~V avutlwav Tiis ypall~S An TTpOS uTTcpvIK1]OIV TOU 16lou ~CtpOUS rlva

bull Ab= f gydx

o

3 To fpyov TOU qopTlou S A8 rlva laov

A ~ S

H tTTlltlKuvaS Tiis ypoliliiiS 11 riVal -lOT) TIpes

1 bull bullt1 = 2 f ydx = I ~I o

H Ct~OVK~ IiUvali~ P riVal tOT) TTp6~ t1

P = E F - shy1

4 To lpyov Tiis ~ov~S liuvalipoundw~ Pv Iia ~v tTTllitlKUVOlV Tiis ypalili~~ Av rlva laoVTTp6S

5 To fpyov Tiis avomuaaolitv]~ 8AmKijs liuvCtlicwS Pp Ap riVal iaov TTp6~

f l 7t1

Ap = Pot1 = PO 4T

MCTa~alvovTcS lit poundIS looppoTTlav liuvCtIipoundwV ixolipoundv litll

S = ampAb = 2EJI ~ b amp1 I

ampAb Sh=I= gT

ampAS8 = tif = S

~v KCtTWa ~Iawalv Sb + Sb+Sv= S +Sp

Kal

= I + ( - P 2fft) + EJ -()S g 2 2EJ T 8- -I _

757 lJl~1 IDlITIIWV 5LaDXOVIKWV x

T Sp f

LX 5 AUYlC1OC rile ypafJr ICOTa TO I(QTaK6pu+o tntm6oY

( ETTEVtpyouaal 6UV6~LC)

f1pound1iofUvoU on ( rII ) t1VQl fllKpOV TO (f II ) WS ClTTelpOaTOV avwTtpas Tll~(wS tmTptshyITel nJV ITapllMlIVIV TOU opOU Sv

OUTW

Kat TO PChOS f bull

213 ( S--f)

f = (hEJ - Pol)

A bullbull df 0oUla Imln u(OV (j[ =

Kat tav S = 0 ITpOKUmtl

Kat

f11Q D(DOflCVOV TPOITOV tmDofliis ~TOI g Kal J yvwara Kat D(DOfltvlV Dlacjgtopav 8(PflOKpashyalas f1t ijTOl P yvwarov haflpllvOfl(V nJV aUVlarwaav ouvafllV S WS auvapTlOlV TWV I Katl

OUTW ITaparWfl(V ypacjgtIKWS nJv WS Iivw auvapTlOlV xapaaaOVTts nJV ITpoundppnAhOUaaV TWV auvapnJapoundwv S = F ( r) (Dla Dlacjgtopa 1 )

OUTW DIll Tiis KaflITUhlS TauTlS tiS TOllS i~ovas auvTtTaYfltvwv SKat f Duvafl(eO va ITpoaDloplawflpoundV TO aITQlTOUflpoundVOV PthOS KOfltjJ(WS TO OIToiov tv auvDuaaflltlgt fit nJV aUVlshyarwaav ouvafllV S ouvaTOl v afToPil tmKlvDUVOV Dla nJv tmDofl~v flOS ~TOI va fTPOKahpound01J hUYlaflov Tiis ypoflfliis KOTa TO KaTOKOpucjgtOV poundfTlfTpoundDOV

H TOlOUTl aITpoundIKOVIOlS Tiis auvoPnlapoundws S = F (g J Po Imiddotf) tmTptfTpound1 nJV aUYKpav DlOcjgt6pwv TUITWV tmDofliis (DlOcjgtOpOl TUITOI OlDlpOTpOX~)v J DlOcjgtOpOl TUITOl arpwshynlPWV g) fIpoundTO~U TWV Kelt TOV IToplaflov aUfllTpoundpoaflOrwv tITt Tiis aacjgtahdos EKaarou TUTTOU

nplnTWal~ II)

IlDlPOTPOXla TUITOU S 49 J =2 x 1781 em E =21 X 1000kp em ITpwTijppounds ~UhlVOl gk 26 = P lem f1lQcjgtopa epoundPfloKpaalos f1t = 45deg C Dl fjv avomUaatTOl a~ovl~

8hlmlK~ ouvofllS Po = 140 Mp

757 lJl~1 IDlITIIWV 5LaDXOVIKWV x

T Sp f

LX 5 AUYlC1OC rile ypafJr ICOTa TO I(QTaK6pu+o tntm6oY

( ETTEVtpyouaal 6UV6~LC)

f1pound1iofUvoU on ( rII ) t1VQl fllKpOV TO (f II ) WS ClTTelpOaTOV avwTtpas Tll~(wS tmTptshyITel nJV ITapllMlIVIV TOU opOU Sv

OUTW

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213 ( S--f)

f = (hEJ - Pol)

A bullbull df 0oUla Imln u(OV (j[ =

Kat tav S = 0 ITpOKUmtl

Kat

f11Q D(DOflCVOV TPOITOV tmDofliis ~TOI g Kal J yvwara Kat D(DOfltvlV Dlacjgtopav 8(PflOKpashyalas f1t ijTOl P yvwarov haflpllvOfl(V nJV aUVlarwaav ouvafllV S WS auvapTlOlV TWV I Katl

OUTW ITaparWfl(V ypacjgtIKWS nJv WS Iivw auvapTlOlV xapaaaOVTts nJV ITpoundppnAhOUaaV TWV auvapnJapoundwv S = F ( r) (Dla Dlacjgtopa 1 )

OUTW DIll Tiis KaflITUhlS TauTlS tiS TOllS i~ovas auvTtTaYfltvwv SKat f Duvafl(eO va ITpoaDloplawflpoundV TO aITQlTOUflpoundVOV PthOS KOfltjJ(WS TO OIToiov tv auvDuaaflltlgt fit nJV aUVlshyarwaav ouvafllV S ouvaTOl v afToPil tmKlvDUVOV Dla nJv tmDofl~v flOS ~TOI va fTPOKahpound01J hUYlaflov Tiis ypoflfliis KOTa TO KaTOKOpucjgtOV poundfTlfTpoundDOV

H TOlOUTl aITpoundIKOVIOlS Tiis auvoPnlapoundws S = F (g J Po Imiddotf) tmTptfTpound1 nJV aUYKpav DlOcjgt6pwv TUITWV tmDofliis (DlOcjgtOpOl TUITOI OlDlpOTpOX~)v J DlOcjgtOpOl TUITOl arpwshynlPWV g) fIpoundTO~U TWV Kelt TOV IToplaflov aUfllTpoundpoaflOrwv tITt Tiis aacjgtahdos EKaarou TUTTOU

nplnTWal~ II)

IlDlPOTPOXla TUITOU S 49 J =2 x 1781 em E =21 X 1000kp em ITpwTijppounds ~UhlVOl gk 26 = P lem f1lQcjgtopa epoundPfloKpaalos f1t = 45deg C Dl fjv avomUaatTOl a~ovl~

8hlmlK~ ouvofllS Po = 140 Mp

758 AvTlKaBOTwvTt~ poundl~ TOV TlmOV

I ( t)S=gT+ 2EJl-Po 21 r

Ka I poundl~ m f poundl~ em

k= 2Eh

(vBa

k=2 a~~Qvof1poundV k = 013 k = 1460 k = 00494

L Po = 140 Mp ~ pound1[awaL~ Aal~QVpoundL T~V Iopcjgt~v

S - 0 13 1+( ~460 _ 69)r- I 1

dfL dT Aal~6vo~poundv I = 1455 m Ka S = 189

H ypacjgtK~ llap6OTaaL~ rij~ auvaprijaw~ 6[6ETClL tv aX~laTL 6

ncpiTrTWOII la

l6~poTpoxa TUllOU S 49 J = 2 x 1781 em E= 2 I x 10 kpf em lTpWr~ppound~ 1 UAVO g = 26 kp fe rn Lacjgtopa epoundp~oKpaa[a~ Lt ~ 60 C 6 ~v 6vaTTTuaana a1ovKL) eATTTK~ Mval~ Po = 200 Mp AVTLKaeOTWVTpound~ d~ TOV TUllOV

S = kl+ n -kto )r

k = 013 k = 1460 Ka k = 00494

Kai 6 Po = 200 Mp Ka kPo = 988

~ t1iawa~ Aal~6vpoundL T~V I0pcjgt~v

S=013+( 1460 _ 988)1 l 1

Lul ~ = 0 Aal~6vol(V 1 = 1215 m Ka S ~ 158

H ypacjgtK~ 1Tap6OTaa~ rij~ auvaprf]apoundw~ 616ETa tv ax 6

CphTTWaIC 3

l6~poTpoxa TUllOU S 49 J = 2 x 1781 em E = 2 1 x 10 kp fern lTPWrijplt~ tK aKupo6tlaTO~ g = 48 kp fern Lacjgtopci ecp~oKpaaia~ L t = lt45 6 ~v avaTTTuaaETClL 61dvKL) eAITTTLKL) Mva~~ Po = 140 Mp

AVTKaBOTwvTt~ ti~ TOV TUllOV

S = kH n shy kiP)1

k = 024 k = liO Ka k = 00494 Ka 6 P = 140 Mp kP = 69

t1iawaL~ Aal~6vpoundL rf]v ~opcjgtL)v

S = 0241+ (t~o - 6t)1

dt I = 1455 In Kal S = 345

H ypacjgtKL) 1Top6OTaOl~ rij~ auvapTLapoundw~ 6i6poundTa tv aX 6

6ci d = 0

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760 nrpinTW(f~ 4 T-IIIilpoTPOXo T(JlTOU 5 lt49 J = 2 x 1781 em E = 2 I x 10 kp lern ITpWTijptS tK oKupolitflaTos g = 48 kp lem 1111T11WI AIQ4gtopo 9tpfloKpaoias At ~ 600 iii ~v avamuootTal a~ovIK~ pnXlVll1IY 9Alm1K~ IiUvafllS Po = 200 Mp AvTlKa91oTWVTtg tiS TOV TI)TTOV

5 - k I + (~ _ ~Po )1- I I

k = 02lt4 k = 1460 Ka k = 00494

Ka lila Po = 200 Mp k Po = 988

l~lowOIS Aaflp4vEI T~V floP4gt~v

5 - 0241 + (1 460 _ 988 )1- Imiddot I

dllila CiT = 0 AaflpavoflEv I = 12 15m Ka 5 = 292

H ypa4gtIK~ TTapaOTaolS T~S ouvapn]oEws IillitTal lv aX 6

nrphwO~ 5 I IIilPOTPOXIO TlJlTOU 5 6lt4 J = 2 3260 em E = 2 I x 10 k p lem I TPWTijpES ~UAIVOI g = 29 kp lem Ala4gtopo 9EPfloKpaoas At = lt450 iii ~v ilvamuooETal O~OVI~

9Alm1K~ li(ivafllt Po = 185 MD AVTlKa910TWVTES tiS TOV TlJlTOV

5= kl + ( ~ - kato )1

k = 0 145 k = 2675 Ka k = 00lt49lt4Ka lila Po = 185 Mo kPo = 915

t~owalS Aaflp4vEI ~v floP4gt~v

5 = 01451 + ( 2675 _ 9i5 )1 I I

dlAla CiT = 0 AaflpavoflEV 1 = 1720 m Kai 5 = 248

H ypa4gtIK~ TTapaOTaOiS Tijs ouvapn]oEws IilliETQI tv AX 7

neuropi1TTWO~ 6 IIIilPOTPOXIO TlJlTOU 5 64 J = 2 X 3260 em E 2 I X 100 kp lem ITpwTijPES tK oKupolifflaTos g r= 51 kp Ir Aa4gtopo aEp~oKpaolas At = lt45deg Ii ~v avamUOOtTala~OVIK~

9Alm1K~ Ii(ivafllS Po = 185 MD AVTIKa9IOTWVTES tiS TOV TITTOV

5 = kl + (~ - ~)I

k = 0255 k = 2675 Ka k = 009lt4

Ka lila Po = 185 Mp kPo = 915

t~ lowolS Aaflp6vEI ~v floP4gt~v

5=O255 1 +C~5 - 9j15 )1

dIlila ill = 0 AaflpavoflEV I = 1720em 5 = lt438

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760 nrpinTW(f~ 4 T-IIIilpoTPOXo T(JlTOU 5 lt49 J = 2 x 1781 em E = 2 I x 10 kp lern ITpWTijptS tK oKupolitflaTos g = 48 kp lem 1111T11WI AIQ4gtopo 9tpfloKpaoias At ~ 600 iii ~v avamuootTal a~ovIK~ pnXlVll1IY 9Alm1K~ IiUvafllS Po = 200 Mp AvTlKa91oTWVTtg tiS TOV TI)TTOV

5 - k I + (~ _ ~Po )1- I I

k = 02lt4 k = 1460 Ka k = 00494

Ka lila Po = 200 Mp k Po = 988

l~lowOIS Aaflp4vEI T~V floP4gt~v

5 - 0241 + (1 460 _ 988 )1- Imiddot I

dllila CiT = 0 AaflpavoflEv I = 12 15m Ka 5 = 292

H ypa4gtIK~ TTapaOTaolS T~S ouvapn]oEws IillitTal lv aX 6

nrphwO~ 5 I IIilPOTPOXIO TlJlTOU 5 6lt4 J = 2 3260 em E = 2 I x 10 k p lem I TPWTijpES ~UAIVOI g = 29 kp lem Ala4gtopo 9EPfloKpaoas At = lt450 iii ~v ilvamuooETal O~OVI~

9Alm1K~ li(ivafllt Po = 185 MD AVTlKa910TWVTES tiS TOV TlJlTOV

5= kl + ( ~ - kato )1

k = 0 145 k = 2675 Ka k = 00lt49lt4Ka lila Po = 185 Mo kPo = 915

t~owalS Aaflp4vEI ~v floP4gt~v

5 = 01451 + ( 2675 _ 9i5 )1 I I

dlAla CiT = 0 AaflpavoflEV 1 = 1720 m Kai 5 = 248

H ypa4gtIK~ TTapaOTaOiS Tijs ouvapn]oEws IilliETQI tv AX 7

neuropi1TTWO~ 6 IIIilPOTPOXIO TlJlTOU 5 64 J = 2 X 3260 em E 2 I X 100 kp lem ITpwTijPES tK oKupolifflaTos g r= 51 kp Ir Aa4gtopo aEp~oKpaolas At = lt45deg Ii ~v avamUOOtTala~OVIK~

9Alm1K~ Ii(ivafllS Po = 185 MD AVTIKa9IOTWVTES tiS TOV TITTOV

5 = kl + (~ - ~)I

k = 0255 k = 2675 Ka k = 009lt4

Ka lila Po = 185 Mp kPo = 915

t~ lowolS Aaflp6vEI ~v floP4gt~v

5=O255 1 +C~5 - 9j15 )1

dIlila ill = 0 AaflpavoflEV I = 1720em 5 = lt438

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5 - k I + (~ _ ~Po )1- I I

k = 02lt4 k = 1460 Ka k = 00494

Ka lila Po = 200 Mp k Po = 988

l~lowOIS Aaflp4vEI T~V floP4gt~v

5 - 0241 + (1 460 _ 988 )1- Imiddot I

dllila CiT = 0 AaflpavoflEv I = 12 15m Ka 5 = 292

H ypa4gtIK~ TTapaOTaolS T~S ouvapn]oEws IillitTal lv aX 6

nrphwO~ 5 I IIilPOTPOXIO TlJlTOU 5 6lt4 J = 2 3260 em E = 2 I x 10 k p lem I TPWTijpES ~UAIVOI g = 29 kp lem Ala4gtopo 9EPfloKpaoas At = lt450 iii ~v ilvamuooETal O~OVI~

9Alm1K~ li(ivafllt Po = 185 MD AVTlKa910TWVTES tiS TOV TlJlTOV

5= kl + ( ~ - kato )1

k = 0 145 k = 2675 Ka k = 00lt49lt4Ka lila Po = 185 Mo kPo = 915

t~owalS Aaflp4vEI ~v floP4gt~v

5 = 01451 + ( 2675 _ 9i5 )1 I I

dlAla CiT = 0 AaflpavoflEV 1 = 1720 m Kai 5 = 248

H ypa4gtIK~ TTapaOTaOiS Tijs ouvapn]oEws IilliETQI tv AX 7

neuropi1TTWO~ 6 IIIilPOTPOXIO TlJlTOU 5 64 J = 2 X 3260 em E 2 I X 100 kp lem ITpwTijPES tK oKupolifflaTos g r= 51 kp Ir Aa4gtopo aEp~oKpaolas At = lt45deg Ii ~v avamUOOtTala~OVIK~

9Alm1K~ Ii(ivafllS Po = 185 MD AVTIKa9IOTWVTES tiS TOV TITTOV

5 = kl + (~ - ~)I

k = 0255 k = 2675 Ka k = 009lt4

Ka lila Po = 185 Mp kPo = 915

t~ lowolS Aaflp6vEI ~v floP4gt~v

5=O255 1 +C~5 - 9j15 )1

dIlila ill = 0 AaflpavoflEV I = 1720em 5 = lt438

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B1IU10ypa+(a - Bibliography

E1lg~l E lgtsterreicliischcs IngenieuJ - Archiv 1960 Tru(o t 2 M tli8 II Organ hil die Fortsctllitta des Eiscnb[l hnwesens middotOxrw~tO 1937 1Iiixo 20 Mricr H E lscnbanlcchnischc Rllndschau 1958 TEUX1) 6 - 7 8 - 9 JFemeUy E Ristnhahntechn Hundschau 1960 I ~uX0 12 R aab F Eisenbahnteehn Rundschau 1958 rdlx~ 11-12 Schra mm G Die E Jscnbahnschione Jahrbuch des EisenbaiLnwosllUs R ohrig - Vlrlag Koln 1 liG S dll4eda F Eistnhahnlcchn Rundschau 1961 T d xo 6 T-fFallrmlTln J LiingskrltlCte in Eiscnbllhngleis E lsner Vcllagsgcscllsrhalt Darmstadt 1957